Deformation quantization of Poisson manifolds, I

Abstract

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.

plain TeX and epsf.tex, 46 pages, 24 figures

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• Noncommutative version of an arbitrary nondegenerated mechanics
• Contravariant Gravity on Poisson Manifolds and Einstein Gravity
• Relational Symplectic Groupoid Quantization for Constant Poisson Structures
• Anomalous Thermal Hall Effect in a Disordered Weyl Ferromagnet
• Multiplicative preprojective algebras are 2-Calabi-Yau
• Deformations of gerbes on smooth manifolds
• Weak associativity and deformation quantization
• Minimal models of quantum homotopy Lie algebras via the BV-formalism
• Locally Noncommutative Space-Times
• The A-infinity Deformation Theory of a Point and the Derived Categories of Local Calabi-Yaus
• Hochschild cohomology of dg manifolds associated to integrable distributions
• Dirac's magnetic monopole and the Kontsevich star product
• Quantization and the tangent groupoid
• Noncommutative relativistic particle
• Gerstenhaber-Batalin-Vilkoviski structures on coisotropic intersections
• Gauge field theories with covariant star-product
• Operads and moduli spaces
• On Globalized Traces for the Poisson Sigma Model
• Quantization of Magnetic Poisson Structures
• Formal oscillatory integrals and deformation quantization
• The Weyl realizations of Lie algebras and left-right duality
• Convergence of the Gutt Star Product
• Deformation quantization of the Pais-Uhlenbeck fourth order oscillator
• Quantization of Poisson manifolds from the integrability of the modular function
• Seiberg-Witten Geometry of Four-Dimensional $\mathcal{N}=2$ Quiver Gauge Theories
• Explicit Formulae for Noncommutative Deformations of CP^N and CH^N
• Non-commutativity parameters depend not only on the effective coordinate but on its T-dual as well
• On higher rank coisotropic A-branes
• Multiplicative structure on the Hochschild cohomology of crossed product algebras
• On the Hochschild-Kostant-Rosenberg map for graded manifolds
• On Coordinate Transformations in Planar Noncommutative Theories
• Aspects of noncommutative descriptions of planar systems in high magnetic fields
• Deformation Quantization of Non Regular Orbits of Compact Lie Groups
• Kontsevich star-product on the dual of a Lie algebra
• Operations on Cyclic Homology, the X Complex, and a Conjecture of Deligne
• Homogeneous Fedosov Star Products on Cotangent Bundles II: GNS Representations, the WKB Expansion, and Applications
• Three Natural Generalizations of Fedosov Quantization
• Noncommutative geometry induced by spin effects
• Comparing Poisson Sigma Model with A-model
• Conjugate variables in quantum field theory: the basic case
• A model for framed configuration spaces of points
• Anomaly cancellation in the topological string
• Realization of associative products in terms of Moyal and tomographic symbols
• Monopole star products are non-alternative
• On beta-deformations and Noncommutativity
• A note on bosonic open strings in constant B field
• Towards the deformation quantization of linearized gravity
• "Quantization is a mystery"
• The graded Lie algebra of general relativity
• Tree-level S-matrix of superstring field theory with homotopy algebra structure
• Bi-metric Universe with Matter
• On the L${}_{\infty }$ structure of Poisson gauge theory
• Isometric Embeddings and Noncommutative Branes in Homogeneous Gravitational Waves
• Wigner functions for angle and orbital angular momentum: Operators and dynamics
• Non-constant Non-commutativity in 2d Field Theories and a New Look at Fuzzy Monopoles
• An Obstruction to Quantization of the Sphere
• Flat family of QFTs and quantization of d-algebras
• Transposed Hom-Poisson and Hom-pre-Lie Poisson algebras and bialgebras
• L-infinity algebras, Cartan homotopies and period maps
• Leibniz rules for enveloping algebras in symmetric ordering
• Boundary G/G theory and topological Poisson-Lie sigma model
• Nilpotent noncommutativity and renormalization
• Characteristic classes of star products on Marsden-Weinstein reduced symplectic manifolds
• Heisenberg double versus deformed derivatives
• A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics
• Cohomology Structures of A Poisson Algebra: II
• Deformation Quantization in Algebraic Geometry
• The Van den Bergh duality and the modular symmetry of a Poisson variety
• Wick Rotations in Deformation Quantization
• Deformation Quantization of Hermitian Vector Bundles
• Cohomology of Feynman graphs and perturbative quantum field theory
• Problem of Time and Background Independence: classical version's higher Lie Theory
• On the emergence of the structure of Physics
• Cyclic Cohomology of the Weyl Algebra
• An algebra of deformation quantization for star-exponentials on complex symplectic manifolds
• Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation
• Einstein-Riemann Gravity on Deformed Spaces
• On a formula of Gammelgaard for Berezin-Toeplitz quantization
• The Universal Generating Function of Analytical Poisson Structures
• The fate of the Wilson-Fisher fixed point in non-commutative ϕ^4
• Quantum backgrounds and QFT
• Gauge Theories in Noncommutative Homogeneous Kähler Manifolds
• Indirect Influences on Directed Manifolds
• Multiplicative McKay correspondence in the symplectic case
• Perturbative Quantum Field Theory and Configuration Space Integrals
• Fermions and noncommutative emergent gravity II: Curved branes in extra dimensions
• Homological epimorphisms, recollements and Hochschild cohomology - with a conjecture by Snashall-Solberg in view
• Graded geometry in gauge theories and beyond
• Notes on noncommutative geometry
• Noncommutative calculus and operads
• Topics on D-brane charges with B-fields
• Poisson-Dirac branes in Poisson-Sigma models
• Covariant star product on symplectic and Poisson spacetime manifolds
• Cohomologies of the Poisson superalgebra on (2,n)-superdimensional spaces
• A Remark on Non-equivalent Star Products via Reduction for CP^n
• Noncommutative theories and general coordinate transformations
• Versality of the relative Fukaya category
• Automorphic Lie algebras and modular forms
• Grothendieck-Teichmüller and Batalin-Vilkovisky
• Phase space quantization, noncommutativity and the gravitational field
• U(2,2) gravity on noncommutative space with symplectic structure
• Nearly associative deformation quantization
• A simple construction of associative deformations
• Fluctuations as stochastic deformation
• Introduction to a Non-Commutative Version of the Standard Model
• Expressions of algebra elements and transcendental noncommutative calculus
• Complete Positivity of Rieffel's Deformation Quantization by Actions of ${\mathbb{R}}^d$
• Homotopical Computations in Quantum Fields Theory
• Twisted submanifolds of R^n
• Moduli problems for operadic algebras
• Hochschild Cohomology versus De Rham Cohomology without Formality Theorems
• On deformations of $A_{\infty }$-algebras
• Review of deformation theory I: Concrete formulas for deformations of algebraic structures
• Topological Dirac Sigma Models and the Classical Master Equation
• The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets
• Formalite $G_{\infty }$ adaptee et star-representations sur des sous-varietes coisotropes
• A note on the Koszul complex in deformation quantization
• Semiquantisation Functor and Poisson-Riemannian Geometry, I
• The Kontsevich graph orientation morphism revisited
• On global deformation quantization in the algebraic case
• On the phase form of a deformation quantization with separation of variables
• Gravity algebra structure on the negative cyclic homology of Calabi-Yau algebras
• Normalized Weyl-type $\star$-product on Kähler manifolds
• A period map for global derived stacks
• A model for complex analytic equivariant elliptic cohomology from quantum field theory
• Field Theory Equivalences as Spans of $L_{\infty }$-algebras
• A N=8 action for multiple M2-branes with an arbitrary number of colors
• Symmetrization of Berezin Star Product and Path-Integral Quantization
• The symbol of a function of an operator
• A Remark on the Deformation of GNS Representations of *-Algebras
• Operadic structure on the Gerstenhaber-Schack complex for prestacks
• Deformation quantization of submanifolds and reductions via Duflo-Kirillov-Kontsevich map
• Effects of wave propagation in canonical Poisson gauge theory under an external magnetic field
• Integrable Hopf twists, marginal deformations and generalised geometry
• Closed constraint algebras and path integrals for loop group actions
• Deformation quantization of constrained systems: a group averaging approach
• Numerical properties of Koszul connections
• Twist star products and Morita equivalence
• Deformation Quantization: Observable Algebras, States and Representation Theory
• Homotopy Theory of Probability Spaces I: Classical independence and homotopy Lie algebras
• Deformations of Azumaya algebras
• The (secret?) homological algebra of the Batalin-Vilkovisky approach
• On the variational noncommutative Poisson geometry
• Centralizers in Free Associative Algebras and Generic Matrices
• Quantisation of derived Poisson structures
• Field Theory on Noncommutative Space-Time and the Deformed Virasoro Algebra
• Algebra of Noncommutative Riemann Surfaces
• Ground-state Wigner functional of linearized gravitational field
• Grothendieck-Teichmueller group, operads and graph complexes: a survey
• Lie algebroids, non-associative structures and non-geometric fluxes
• Star products and branes in Poisson-Sigma models
• Noncommutative quantum mechanics as a constrained system
• Operads, configuration spaces and quantization
• Dynamical Mass Generations and Collective Excitations in the (Supersymmetric-)Nambu$-$Jona-Lasinio Model and a Gauge Theory with Left-Right-Asymmetric Majorana Mass Terms
• Balanced metrics and noncommutative Kaehler geometry
• The deformation quantization mapping of Poisson- to associative structures in field theory
• Constant H field, cosmology and faster than light solitons
• Conserved symmetries in noncommutative quantum mechanics
• Deformation quantization of gerbes
• Sh-Lie algebras Induced by Gauge Transformations
• Continuous and Twisted L_infinity Morphisms
• How to quantize the antibracket
• Formal Lagrangian Operad
• Graph complexes in deformation quantization
• The minimal model for the Batalin-Vilkovisky operad
• Poisson cohomology, Koszul duality, and Batalin-Vilkovisky algebras
• Formality of differential graded algebras and complex Lagrangian submanifolds
• Twisted Fock Representations of Noncommutative Kähler Manifolds
• Homology of formal deformations of proper etale Lie groupoids
• On Computational Poisson Geometry II: Numerical Methods
• A Universal Construction of Universal Deformation Formulas, Drinfel'd Twists and their Positivity
• Growth of nonsymmetric operads
• $L_{\infty }$ and $A_{\infty }$ structures: then and now
• Comparison and Continuity of Wick-type Star Products on certain coadjoint orbits
• Generating functions for local symplectic groupoids and non-perturbative semiclassical quantization
• A combinatorial approach to coefficients in deformation quantization
• Hopf algebras arising from dg manifolds
• The Gerstenhaber bracket as a Schouten bracket for polynomial rings extended by finite groups
• Invariant Hochschild cohomology of smooth functions
• Nijenhuis infinity and contractible dg manifolds
• Polyvector fields for Fano 3-folds
• Higher Koszul brackets on the cotangent complex
• Batalin--Vilkovisky quantization and supersymmetric twists
• On deformation theory and graph homology
• Trace functionals on non-commutative deformations of moduli spaces of flat connections
• Generalized dilaton gravity in 2d
• Noncommutative Reissner-Nordstrøm Black hole
• Lifshitz field theories, Snyder noncomutative space-time and momentum dependent metric
• A Hopf algebra deformation approach to renormalization
• The role of self-induced decoherence in the problem of the classical limit of quantum mechanics
• Lagrangian and Noncommutativity
• Dixmier traces on noncompact isospectral deformations
• Revisiting NCQED and scattering amplitudes
• Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs
• Nijenhuis-type variants of Local Theory of Background Independence
• Non-Abelian Lie algebroids over jet spaces
• Deformations of linear Poisson orbifolds
• Shoikhet's Conjecture and Duflo Isomorphism on (Co)Invariants
• Homotopy transfer and rational models for mapping spaces
• On the Spectral Sequence of the Swiss-cheese Operad
• Weyl invariant polynomial and deformation quantization on Kahler manifolds
• Lorentz invariant supersymmetric mechanism for non(anti)commutative deformations of space-time geometry
• Poisson electrodynamics on $\kappa$-Minkowski space-time
• Shifted Poisson structures on differentiable stacks
• Deformation Quantization of Surjective Submersions and Principal Fibre Bundles
• Target Space Equivariant Cohomological Structure of the Poisson Sigma Model
• Noncommutative field theories from a deformation point of view
• Poisson Sigma Model with branes and hyperelliptic Riemann surfaces
• Poisson deformations of symplectic quotient singularities
• Non-commutativity and Open Strings Dynamics in Melvin Universes
• Quantum twistors
• An invariant formula for a star product with separation of variables
• A General formulation of the Moyal and Voros products and its physical interpretation
• Factorization Algebras for Bulk-Boundary Systems
• Fréchet Quantum Supergroups
• Enveloping algebra Noncommutative SM: Renormalisability and High Energy Physics Phenomenology
• On the isomorphism between the reduction algebra and the invariant differential operators on Lie groups
• BRST quantization of quasi-symplectic manifolds and beyond
• Algebraic Principles of Quantum Field Theory I: Foundation and an exact solution of BV QFT
• NC Wilson lines and the inverse Seiberg-Witten map for nondegenerate star products
• Invariant star-products on S^2 and the canonical trace
• The Fuzzy Sphere: From The Uncertainty Relation To The Stereographic Projection
• Branching random walk solutions to the Wigner equation
• The Hierarchy Problem: From the Fundamentals to the Frontiers
• A short note on infinity-groupoids and the period map for projective manifolds
• Deformations of path algebras of quivers with relations
• Convergence of Star Product: From Examples to a General Framework
• Noncommutative Lagrange Mechanics
• Differential Poisson Sigma Models with Extended Supersymmetry
• Massless chiral fields in six dimensions
• Matrix Models, Emergent Gravity, and Gauge Theory
• Lagrangian field theories: ind/pro-approach and L-infinity algebra of local observables
• Algebras, traces, and boundary correlators in $\mathcal{N}=4$ SYM
• A note on a question of Markman
• Deformations of vector bundles on coisotropic subvarieties via the Atiyah class
• On Gammelgaard's formula for a star product with separation of variables
• On L-infinity morphisms of cyclic chains
• Algebraic Principles of Quantum Field Theory II: Quantum Coordinates and WDVV Equation
• Categories of holomorphic line bundles on higher dimensional noncommutative complex tori
• MC Elements in Pronilpotent DG Lie Algebras
• Tautological Tuning of the Kostant-Souriau Quantization Map with Differential Geometric Structures
• Dark Matter in a bi-metric universe
• Relative Recognition Principle
• Deformation quantization and the tomographic representation of quantum fields
• On the q-meromorphic Weyl algebra
• Dunkl operator and quantization of ${\mathbb{Z}}_2$-singularity
• Canonical Functions and Differential Graded Symplectic Pairs in Supergeometry and AKSZ Sigma Models with Boundary
• Violation of the phase space general covariance as a diffeomorphism anomaly in quantum mechanics
• Hilbert space formalisms for group field theory
• Matrix Configurations for Spherical 4-branes and Non-commutative Structures on S^4
• Canonical Deformation of $N=2$ $Ad{S}_4$ SUGRA
• The global derived period map
• Formal Global AKSZ Gauge Observables and Generalized Wilson Surfaces
• Configuration Spaces of Manifolds with Boundary
• Weakly associative algebras, Poisson algebras and quantization
• BRST Reduction of Quantum Algebras with ${}^{\ast }$-Involutions
• Augmented Polynomial Symplectomorphisms and Quantization
• Nambu variant of Local Resolution of Problem of Time and Background Independence
• Remarks on derived equivalences of Ricci-flat manifolds
• Algebroid prestacks and deformations of ringed spaces
• The big Chern classes and the Chern character
• Canonical aspects of pregeometric vector-based first order gauge theory
• Branes and Quantization for an A-Model Complexification of Einstein Gravity in Almost Kahler Variables
• A relative version of the ordinary perturbation lemma
• Formality theorem for quantizations of Lie bialgebras
• Globalizing L-infinity automorphisms of the Schouten algebra of polyvector fields
• Deformation Quantization of Symplectic Fibrations
• H-äquivariante Morita-Äquivalenz und Deformationsquantisierung
• On the inverse mapping of the formal symplectic groupoid of a deformation quantization
• Twisted Quadrics and Algebraic Submanifolds in ${\mathbb{R}}^n$
• Open problems in the Kontsevich graph construction of Poisson bracket symmetries
• Matter-gravity coupling for fuzzy geometry and the Landau-Hall problem
• Keller admissible triples and Duflo theorem
• Symmetric quantum Weyl algebras
• A Deformation Quantization for Non-Flat Spacetimes and Applications to QFT
• $L_{i}nfty$ rational homotopy of mapping spaces
• Extensions of the Duflo map and Chern-Simons expectation values
• Higher algebraic structures in Hamiltonian Floer theory
• L-infinity Formality check for the Hochschild Complex of Certain Universal Enveloping Algebras
• On the compatibility between cup products, the Alekseev--Torossian connection and the Kashiwara--Vergne conjecture
• Star Product Reduction for Coisotropic Submanifolds of Codimension 1
• Path algebras, wave-particle duality, and quantization of phase space
• The right-hand side of the Jacobi identity: to be naught or not to be?
• Derived geometry of the first formal neighborhood of a smooth analytic cycle
• A Counter Example of Invariant Deformation Quantization
• 2D Poisson Sigma Models with Gauged Vectorial Supersymmetry
• Quantum Theory, Noncommutativity and Heuristics
• Nonabelian mixed Hodge structures
• Noncommutative gerbes and deformation quantization
• Weyl-Underhill-Emmrich quantization and the Stratonovich-Weyl quantizer
• Identification of Berezin-Toeplitz deformation quantization
• Deligne groupoid revisited
• The explicit equivalence between the standard and the logarithmic star product for Lie algebras
• Supersymmetric field theories and the elliptic index theorem with complex coefficients
• Constants of Motion of the Harmonic Oscillator
• Strict quantizations of almost Poisson manifolds
• Discrete Wigner-Weyl calculus for the finite lattice
• Lectures on Deformation quantization of Poisson manifolds
• The expansion $\star$ mod $\bar{o}({\hslash }^4)$ and computer-assisted proof schemes in the Kontsevich deformation quantization
• Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism
• Presentations of non-commutative deformation rings via $A_{\infty }$-algebras and applications to deformations of Galois representations and pseudorepresentations
• Almost Kaehler deformation quantization
• On the Kontsevich $\star$-product associativity mechanism
• The Anomalous Nambu-Goldstone Theorem in Relativistic/Nonrelativistic Quantum Field Theory
• Moyal-Weyl Star-products as Quasiconformal Mappings
• Operad profiles of Nijenhuis structures
• Branes and DAHA Representations
• Polyvector fields and polydifferential operators associated with Lie pairs
• On representations of star product algebras over cotangent spaces on Hermitian line bundles
• Categorical Perspective on Quantization of Poisson Algebra
• A covariant Poisson deformation quantization with separation of variables up to the third order
• Algebraic models of non-connected spaces and homotopy theory of $L_{\infty }$ algebras
• Deformation quantization of a Kaehler-Poisson structure vanishing on a Levi nondegenerate hypersurface
• Deformation quantization of Leibniz algebras
• Atiyah classes of strongly homotopy Lie pairs
• Characteristic classes in deformation quantization
• Formality theorem for Lie bialgebras and quantization of coboundary r-matrices
• Associative and Lie deformations of Poisson algebras
• On the large N limit, W_\infty Strings, Star products, AdS/CFT Duality, Nonlinear Sigma Models on AdS spaces and Chern-Simons p-branes
• $\kappa$-General-Relativity I: a Non-Commutative GR Theory with the $\kappa$-Minkowski Spacetime as its Flat Limit
• Deformation quantization of the simplest Poisson Orbifold
• A category of kernels for equivariant factorizations, II: further implications
• On topology changes in quantum field theory and quantum gravity
• Nijenhuis forms on $L_{\infty }$-algebras and Poisson geometry
• Quadratic and pinczon algebras
• Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras
• Extension, deformation and categorification of $\text{AssDer}$ pairs
• Homological sections of Lie algebroids
• On the relation between geometric and deformation quantization
• Deformation Quantization of a Certain Type of Open Systems
• Field Theory with Coordinate Dependent Noncommutativity
• Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems
• On the (co)homology of Frobenius Poisson algebras
• Contact Quantization: Quantum Mechanics = Parallel transport
• Infinitesimal braidings and pre-Cartier bialgebras
• A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra
• On coisotropic deformations of holomorphic submanifolds
• Formal symplectic realizations
• Versality in mirror symmetry
• Regularized Integrals on Riemann Surfaces and Modular Forms
• Deformations of anti-de Sitter black holes
• Quantized Equations of Motion and Currents in Noncommutative Theories
• Homotopy graph-complex for configuration and knot spaces
• Symplectic Microgeometry IV: Quantization
• The Ext algebra of a quantized cycle
• Combinatorial proof of a Non-Renormalization Theorem
• Kontsevich graphs act on Nambu-Poisson brackets, I. New identities for Jacobian determinants
• Split Canonical Relations
• Equivariant quantization of Poisson homogeneous spaces and Kostant's problem
• Hochschild Homology and Cohomology of Klein Surfaces
• Translation-Invariant Noncommutative Gauge Theories, Matrix Modeling and Noncommutative Geometry
• A proof of Tsygan's formality conjecture for Hamiltonian actions
• Quantization of $r-Z$-quasi-Poisson manifolds and related modified classical dynamical $r$-matrices
• Configuration-mapping spaces and homology stability
• Algorithmic deformation of matrix factorisations
• Global Space-Time Symmetries of Quantized Euclidean and Minkowski Superspaces
• Operations on the Hochschild Bicomplex of a Diagram of Algebras
• An invitation to algebraic topological string theory
• Operad of formal homogeneous spaces and Bernoulli numbers
• Superfiltered $A_{\infty }$-deformations of the exterior algebra, and local mirror symmetry
• A sheaf of Hochschild complexes on quasi-compact opens
• On Nonlinear Gauge Theories
• Deformation Expression for Elements of Algebras (VI) --Vacuum representation of Heisenberg algebra--
• Eulerian idempotent and Kashiwara-Vergne conjecture
• A note on Q-algebra and quantization
• Deformation of line bundles on coisotropic subvarieties
• On the concept of a filtered bundle
• Quantum spaces, central extensions of Lie groups and related quantum field theories
• Deformations of Levi-flat structures in smooth manifolds
• Infinity Structure of Poincare Duality Spaces
• Formality for algebroid stacks
• On deformation theory of quantum vertex algebras
• From topological field theory to deformation quantization and reduction
• On the Correspondence of Open and Closed Strings
• An explicit two step quantization of Poisson structures and Lie bialgebras
• The Homotopy Class of twisted $L_{\infty }$-morphisms
• Infinitesimal deformations of a formal symplectic groupoid
• Dual branes in topological sigma models over Lie groups. BF-theory and non-factorizable Lie bialgebras
• Quantum Dynamics on the three-dimensional q-deformed Euclidean Space
• Floer cohomology and pencils of quadrics
• Homotopy relative Rota-Baxter Lie algebras, triangular $L_{\infty }$-bialgebras and higher derived brackets
• A geometric construction of representations of the Berezin-Toeplitz quantization
• Projective module description of embedded noncommutative spaces
• The deformations of nondegenerate constant Poisson bracket with even and odd deformation parameters
• Emergent geometry of membranes
• On the universality of string theory
• Deformations of overconvergent isocrystals on the projective line
• Algebraic and Differential Star Products on Regular Orbits of Compact Lie Groups
• Quantization of pencils with a gl-type Poisson center and braided geometry
• Calculus of multilinear differential operators, operator $L_{\infty }$-algebras and ${IBL}_{\infty }$-algebras
• Quantum symmetric functions
• 4-Manifold Topology, Donaldson-Witten Theory, Floer Homology and Higher Gauge Theory Methods in the BV-BFV Formalism
• Nonperturbative effects in deformation quantization
• Cyclic operad formality for compactified moduli spaces of genus zero surfaces
• Noncommutative Supergeometry and Quantum Supergroups
• Wilson RG of Noncommutative ${\Phi }_4^4$
• The general form of the star-product on the Grassman algebra
• Lectures on Strings, D-branes and Gauge Theories
• Formality of a higher-codimensional Swiss-Cheese operad
• Some Remarks on {$\mathfrak{g}$}-invariant Fedosov Star Products and Quantum Momentum Mappings
• Non Standard Extended Noncommutativity of Coordinates
• The Moyal Momentum Algebra
• Frobenius_infinity invariants of homotopy Gerstenhaber algebras I
• Quantum Systems Exactly
• Explicit Homotopy Equivalences Between Some Operads
• Derived deformation theory of algebraic structures
• Poisson sigma model and semiclassical quantization of integrable systems
• A finiteness theorem for holonomic DQ-modules on Poisson manifolds
• New Universal Deformation Formulas for deformation quantization
• A holography theory of Poisson sigma model and deformation quantization
• A construction of an iterated Ore extension
• Towards deformation quantization over a Z-graded base
• Quantum moment map and obstructions to the existence of closed Fedosov star products
• Abstract Dynamical Systems: Remarks on Symmetries and Reduction
• A Geometric Approach to Hochschild Cohomology of the Exterior Algebra
• Double Poisson extensions
• Homotopy theory of $C_{\infty }$-algebras and characteristic classes of fiber bundles
• Alternative multiplications and non-associativity in physics
• The Quantum Darboux Theorem,
• Connections and $L_{\infty }$ liftings of semiregularity maps
• The Atiyah class of generalized holomorphic vector bundles
• The standard electroweak model in the noncommutative $DFR$ space-time
• Frobenius Manifold Structure on Dolbeault Cohomology and Mirror Symmetry
• Identification of Two Frobenius Manifolds In Mirror Symmetry
• Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures
• Canonical Quantization of Constants of Motion
• Holographic Entanglement Entropy for noncommutative Anti-de Sitter space
• Quantization in Cartesian coordinates and the Hofer metric
• Calabi-Yau algebras viewed as deformations of Poisson algebras
• The Grothendieck-Teichmüller group action on differential forms and formality morphism of chains
• Deformation quantization with separation of variables of an endomorphism bundle
• Normal forms of $\mathbb{Z}$-graded $Q$-manifolds
• Groupoids, Frobenius algebras and Poisson sigma models
• On local equivalence of star-products on Poisson manifolds
• A gentle introduction to Drinfel'd associators
• Quantum mechanics from invariance principles
• On higher order Leibniz identities in TCFT
• The noncommutative Poisson bracket and the deformation of the family algebras
• The Dolbeault dga of the formal neighborhood of the diagonal
• Deformation Quantization via Categorical Factorization Homology
• Consistent deformations in the presymplectic BV-AKSZ approach
• Differentiable groupoid objects and their abstract Lie algebroids
• Threefold Nature of Graded Vector Bundles
• Deformations and homotopy of Rota-Baxter operators and $\mathcal{O}$-operators on Lie algebras
• A noncommutative geometric approach to the quantum structure of spacetime
• Equivalence theorems and an explicit formula for the biquantization character
• The cup product in orbifold Hochschild cohomology
• Filtered expansions in general relativity II
• Weyl n-algebras and the Swiss cheese operad
• Quantum Field Theory and the Volume Conjecture
• Homologies of monomial operads and algebras
• Structures of BiHom-Poisson algebras
• $L_{\infty }$-derivations and the argument shift method for deformation quantization algebras
• Braided Commutative Geometry and Drinfel'd Twist Deformations
• The $L_{\infty }$-algebra of the S-matrix
• Morphisms, direct sums and tensor products of Kähler-Poisson algebras
• $L_{\infty }$-actions of Lie algebroids
• Kähler-Poisson algebras
• The coupling of Poisson sigma models to topological backgrounds
• Deformation quantization and the action of Poisson vector fields
• On the Lie 2-algebra of sections of an LA-groupoid
• Complex manifolds as families of homotopy algebras
• Multipliers of Hilbert algebras and deformation quantization
• Dirichlet branes and a cohomological definition of time flow
• Decomplexification of the Capelli identities and holomorphic factorization
• Poisson sigma models on surfaces with boundary: classical and quantum aspects
• Grothendieck-Teichmueller group and Poisson cohomologies
• Conservation law in noncommutative geometry -- Application to spin-orbit coupled systems
• Marginal Deformations of N=4 SYM and Open vs. Closed String Parameters
• Symbol calculus on a projective space
• All Stable Characteristic Classes of Homological Vector Fields
• D-Branes in Noncommutative Field Theory
• Quantum mechanics and classical trajectories
• A note on the implementation of Poincaré symmetry in noncommutative field theory
• Wrapping Brownian motion and heat kernels II: symmetric spaces
• Microlagrangian manifolds and quasithermodynamic fluctuations of nonequilibrium states
• Quantum gauge fields and flat connections in 2-dimensional BF theory
• Geometric quantization and non-perturbative Poisson sigma model
• Noncommutative 6D Gauge Higgs Unification Models
• Spectrum of the harmonic oscillator in a general noncommutative phase space
• Retarded Functions in Noncommutative Theories
• Noncommutative Poisson structures on Orbifolds
• Natural differential operations on manifolds: an algebraic approach
• Deformation Quantization and Quaternions
• Convergence of the Wick Star Product
• Quantum Dynamics on the Worldvolume from Classical su(n) Cohomology
• Deformations of algebroid stacks
• Geometric and deformation quantization
• Grassmann Electrodynamics and General Relativity
• Lectures on Noncommutative Geometry
• Deformation Quantization of Relativistic Particles in Electromagnetic Fields
• The Kontsevich weight of a wheel with spokes pointing outward
• A universal symmetry structure in open string theory
• Prop profile of bi-Hamiltonian structures
• Morita equivalence of Fedosov star products and deformed Hermitian vector bundles
• Generalisations of Rozansky-Witten invariants
• A review on deformation quantization of coadjoint orbits of semisimple Lie groups
• On a noncommutative deformation of the Connes-Kreimer algebra
• The Structure of Spacetime and Noncommutative Geometry
• Deformation of a smooth Deligne-Mumford stack via differential graded Lie algebra
• A Lie-Algebra model for a noncommutative space time geometry
• On the deformation quantization of symplectic orbispaces
• Covariant and Equivariant Formality Theorems
• The Seiberg-Witten Map on the Fuzzy Sphere
• Gravitational Anomalies in Noncommutative Field Theory
• A Simple Algebraic Proof of the Algebraic Index Theorem
• Cobordism of symplectic manifolds and asymptotic expansions
• Quantum Mechanics and Operator algebras on the Hilbert ball
• The universal envelope of the topological closed string BRST-complex
• Polydiagonal compactification of configuration spaces
• High Energy Commutators in Particle, String and Membrane Theories
• Multisymplectic Structures and Higher Momentum Maps
• Notes on Derived Geometric Formulations in Physics
• Twisted Weil Algebras for the String Lie 2-Algebra
• Noncommutative approach to diagnose degenerate Higgs bosons at 125 GeV
• The $L_{\infty }$-structure on symplectic cohomology
• Cohomology Groups of Deformations of Line Bundles on Complex Tori
• Distortion of the Poisson Bracket by the Noncommutative Planck Constants
• Polynomial automorphisms, quantization and Jacobian conjecture related problems
• Momentum and Position Representations for the q-deformed Euclidean Quantum Space
• Lifting central invariants of quantized Hamiltonian actions
• Non-abelian D=11 Supermembrane
• Representations of fermionic star product algebras and the projectively flat connection
• On the importance of being primitive
• Non-commutative Poisson Structures on quantum torus orbifolds
• Dynamical Quantization of Contact Structures
• Generalizations of Schouten-Nijenhuis Bracket
• On the L-infinity description of the Hitchin Map
• Conservation laws for a $q$-deformed nonrelativistic particle
• Noncommutative localization in smooth deformation quantization
• Lectures on Symplectic Geometry, Poisson Geometry, Deformation Quantization and Quantum Field Theory
• Cohomologie De Hochschild Des Surfaces De Klein
• $L_{\infty }$-algebra of an unobstructed deformation functor
• Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology
• On the computation of star products
• Bulk deformations of open topological string theory
• Covariant Strong Morita Theory of Star Product Algebras
• Nonrelativistic one-particle problem on $q$-deformed Euclidean space
• Rarita-Schwinger Quantum Free Field Via Deformation Quantization
• The Chevalley--Eilenberg complex and deformation quantization in presence of two branes
• Quantum moment maps and symmetric bounded domains quantizations
• Associative algebras, punctured disks and the quantization of Poisson manifolds
• Une vue panoramique sur l'analyse semi-classique
• Quantum Space-Time and Noncommutative Gauge Field Theories
• Gradient expansion of the non-Abelian gauge-covariant Moyal star-product
• Posetted trees and Baker-Campbell-Hausdorff product
• Quantum maps and automorphisms
• Cohomology of morphism Lie algebras and some applications
• Star products on generalised complex manifolds
• Quantizing Derived Mapping Stacks
• Quantum Lichnerowicz - Poisson complex
• A Survey of Huebschmann and Stasheff's Paper: Formal Solution of the Master Equation via HPT and Deformation Theory
• Two-dimensional non-abelian BF theory in Lorenz gauge as a solvable logarithmic TCFT
• Lift of $C\_\infty$ and $L\_\infty$ morphisms to $G\_\infty$ morphisms
• R-Matrix Poisson Algebras and Their Deformations
• Maurer-Cartan moduli and models for function spaces
• Snyder and their representation with creation and annihilation operators
• Holomorphic field theories and higher algebra
• Erratum to: Deformation Quantization in Algebraic Geometry
• An Averaging Process for Unipotent Group Actions
• Operator Ordering in the Relativistic Quantization: Specific Heat in the Rindler Frame
• Exhausting formal quantization procedures
• Some Consequences of the Grunewald-O'Halloran Conjecture for Pseudoquonic Operators
• Loop Corrected Supercharges from Holomorphic Anomalies
• Cohomologies of superalgebra of pointwise superproduct
• Derived operations satisfy standard identities
• Cotangent Microbundle Category, I
• On Schwarzschild black holes in a D-dimensional noncommutative space
• Supersymmetric Poisson and Poisson-supersymmetric sigma models
• N=1/2 Global SUSY: R-Matrix Approach
• The homology theory of Koszul-Vinberg algebroids and Poisson manifolds II
• From Kontsevich Graphs to Feynman graphs, a Viewpoint from the Star Products of Scalar Fields
• The cohomology of Koszul-Vinberg algebras
• Torus actions on free associative algebras, lifting and Białynicki-Birula type theorems
• Convolution algebras for Relational Groupoids and Reduction
• Hopf Algebras Consisting of Finite Sets
• Note on Floer theory and integrable hierarchies
• On induced L-infinity action of diffeomorphisms on Cochains
• Quantum Spacetimes from General Relativity?
• Mixed Resolutions and Simplicial Sections
• On the Lie-formality of Poisson manifolds
• W- algebras and Duflo Isomorphism
• Deformation Quantization and Quantum Field Theory on Curved Spaces: the Case of Two-Sphere
• Poisson (co)homology of truncated polynomial algebras in two variables
• Formal deformations of Poisson structures in low dimensions
• A Graphical Calculus for Classical and Quantum Microformal Morphisms
• A Negative Answer to a Question by Rieffel
• A heat kernel proof of the index theorem for deformation quantization
• Associativity certificates for Kontsevich's star-product $\star$ mod $\bar{o}({\hslash }^k)$: $k⩽6$ unlike $k⩾7$
• $\hslash$-Vertex algebras and chiralization of star products
• Irreducible highest-weight modules and equivariant quantization
• Omni-Lie Color Algebras and Lie Color 2-Algebras
• Geometic vertex operators
• Non-formality of Voronov's Swiss-Cheese operads
• "A combinatorial universal $\star$-product" is incorrect
• A Physical Origin for Singular Support Conditions in Geometric Langlands Theory
• On the abstract Bogomolov-Tian-Todorov Theorem
• Equivariant deformation quantization and coadjoint orbit method
• Instances of Higher Geometry in Field Theory
• The Chern-Gauss-Bonnet Theorem via supersymmetric Euclidean field theories
• Quantization of Whitney functions and reduction
• On the geometry of the characteristic class of a star product on a symplectic manifold
• Chiral Higher Spin Gravity in (A)dS${}_4$ and secrets of Chern-Simons Matter Theories
• Tangential Star Products
• Boardman-Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules
• Explicit formula of deformation quantization with separation of variables for complex two-dimensional locally symmetric Kähler manifold
• Vertex algebras and vertex Poisson algebras
• $\Ughr$ invariant quantization on some homogeneous manifolds
• Quantization of SL(2,R)^* as Bialgebra
• The Strong Homotopy Structure of BRST Reduction
• Construction of Spacetime Field and Quantization of General Relativity
• A symplectic prolegomenon
• Quantum Product of Symmetric Functions
• Strict quantization of polynomial Poisson structures
• The general form of the *-commutator on the Grassman algebra
• Covariant Star Product on Semi-Conformally Flat Noncommutative Calabi-Yau Manifolds and Noncommutative Topological Index Theorem
• Deformations of associative algebras with inner products
• The gauge action, DG Lie algebra and identities for Bernoulli numbers
• Kontsevich's deformation quantization: from Dirac to multiple zeta values
• Semi-infinite A-variations of Hodge structure over extended Kahler cone
• Quadratic tomography star product algebra and its classical limit
• Induction of Hamiltonian Poisson actions
• Versal dg deformation of Calabi--Yau manifolds
• Poisson Diffeomorphism Groups
• Hypersurfaces and generalized deformations
• Causal Poisson bracket via deformation quantization
• The orbifold Hochschild product for Fermat hypersurface
• Atiyah and Todd classes of regular Lie algebroids
• Generalizations for Schouten-Nijenhuis Bracket and for Differential Analog of Special Yang-Baxter Equations
• Deformation Quantization for Supermanifolds via Gelfand-Kazhdan Descent
• On the Dynamics of the Self Gravitating Ellipsoid in N dimensions and its Deformation Quantization
• Admissible Poisson bialgebras
• The AdS^2_θ/CFT_1 Correspondence and Noncommutative Geometry II: Noncommutative Quantum Black Holes
• Computation of Kontsevich Weights of Connection and Curvature Graphs for Symplectic Poisson Structures
• Deformations of Poisson algebras
• Moyal-Weyl deformations of $\mathrm{DGA}$ and $\mathrm{DGCA}$
• Some notes on string theory
• Insertion and Lie Bracket Concerning Finite Sets
• Gravity prop and moduli spaces ${\mathcal{M}}_{g,n}$
• On the quantization of $C^{\infty }({\mathbb{R}}^d)$
• Noncommutative Cohomological Field Theories and Topological Aspects of Matrix models
• A Rational Surgery Formula for the LMO Invariant
• Models for the n-Swiss Cheese operads
• Deformations of categories of coherent sheaves via quivers with relations
• Formal Global Perturbative Quantization of the Rozansky-Witten Model in the BV-BFV Formalism
• Lie-Rinehart algebra $\simeq$ acyclic Lie $\infty$-algebroid
• Superspace Generalizations of Schouten-Nijenhuis Bracket
• On the General Structure of the Non-Abelian Born-Infeld Action
• A Formality Theorem for Hochschild Chains
• A duality Hopf algebra for holomorphic N=1 special geometries
• Quantum supertwistors
• Quantum Mechanics in Noncommutative space
• An alternative way to explain how non-commutativity arises in the bosonic string theory
• K-theory in cutoff version of Vacuum String Field Theory
• Deformation quantization on the cotangent bundle of a Lie group
• A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids
• Deformation quantization on jet manifolds
• Classification of Group Actions Extended to Symplectic Deformation Quantizations
• UV/IR mixing in Noncommutative SU(N) Yang-Mills theory
• A Duflo Star Product for Poisson Groups
• On the deformation quantization of affine algebraic varieties
• Rational homotopy type of mapping spaces via cohomology algebras
• Holomorphically Covariant Matrix Models
• Biquantization techniques for computing characters of differential operators on Lie groups
• Deformation quantization and invariant differential operators
• Mode d'emploi de la théorie constructive des champs bosoniques (A user's guide to bosonic constructive field theory)
• Star products with separation of variables admitting a smooth extension
• Deformation quantization on a Hilbert space
• Stargenfunctions, generally parametrized systems and a causal formulation of phase space quantum mechanics
• D-branes, actions effectives et symétrie miroir
• An ICT based solution to make mines safer
• Cohomologie de Chevalley des graphes ascendants
• On The Formality Theorem for the Differential Graded Lie Algebra of Drinfeld
• Superalgebraic interpretation of quantization maps of Weil algebras
• Deformations of Affine Varieties and the Deligne Crossed Groupoid
• Deformations and D-branes
• Formality theorem with coefficients in a module
• The universality question for noncommutative quantum field theory
• Deformation Quantization of A-infinity Equivalences
• The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal
• Compactified configuration space of points on a line and homotopies of $A_{\infty }$ morphisms
• Hochschild Cohomology and Twisted Complexes on Complex Manifolds
• Singular propagators in deformation quantization and Shoikhet-Tsygan formality
• Non-commutative Gravity and the Einstein-Van der Waals Equation of State
• Algebraic Deformation Quantization
• Characteristic (Fedosov-)class of a twist constructed by Drinfel'd
• Does the fivebrane have a nonclassical BV-structure?
• On the Hochschild homology of elliptic Sklyanin algebras
• Universal enveloping algebras of differential graded Poisson algebras
• Formality theorem for Hochschild (co)chains of the algebra of endomorphisms of a vector bundle
• A formality framework for commutative deformations
• Introduction to finite $W$-algebras
• Star Products that can not be induced by Drinfel'd Twists
• A formal model of Berezin-Toeplitz quantization
• Bosonic Higher Spin Gravity in any Dimension with Dynamical Two-Form
• A noncommutative calculus on the cyclic dual of Ext
• Automorphisms of the Weyl manifold
• Noncommutative Deformations of Locally Symmetric Kähler manifolds
• Deformation quantization in the teaching of Lie group representations
• Graph Reconstruction, Functorial Feynman Rules and Superposition Principles